698
DOC.
659 NOVEMBER
1918
space
(in
which
the
rigid body
would
then
have
to
be in
a
form
of
rest),
the
mentioned conditions
can
be realized.
If
this
is
not
satisfied,
I
do
not
see
how
the
“rigid body” ought
to
be defined
accordingly.-
Then
I
also have
another
question
on
my
mind with
regard
to
the
facts of
Weyl's
theory.[2]
Would it
not
be
possible
that,
instead
of
ƒ /Euvguvdxudxv,
the
“length” were given
by
an
expression
ƒ
/k
•
Euvguvdxudxv,
where
k means an
invariant function
of
x1, x2, x3,
x4
against
coordinate
transformations
expressible
by
guv
and
Qv,
which function
is
constant in
the
case
where
guv
=
6uv
and
which,
upon
introduction
of
Weyl’s
A-factor, changes
into const
.
A-1
.
k?
Then,
of
course,
all
length
relations would be invariants in
Weyl's sense,
and
we
would have
the
usual
length
measurement
for
Euclidean
metrics.[3]
Given,
for
inst.,
the
world
were
composed
in such
a
way
that
in it
a
point O
could be chosen
so
that the
geodesic
lines
starting
from
O
covered
the
world
com-
pletely
and
simply
(except
for
O
itself,
of
course),
then the
expression
e-
fpo
EvQvdxv
in which
the
integral
extends
over
the
geodesic
link between O and P,
would be
a
function
k
of
the
coordinates
of P, which has
the
required properties.[4]
Hilbert,
from whom
the
idea
of
the introduction
of
a k
factor
originates,
has
indicated the
equation
i/rE/'-i'
H
\
--
1
dn
k
dx.
=
0
to
determine
such
a
function[5] which,
interpreted
as a
differential equation for
the
k
to
be
determined,
is
invariant
against
coordinate
transformations
and,
in
addition,
remains
unchanged
if
guv
is
replaced by
A
•
guv,
Qv
by Qv+
1dA/Adxv,
and
k
by
A-1
•
k.
Perhaps through
invariant
constraints
a
solution
dependent
on
the
guv's
and
Qv's
could be chosen for
this
differential
equation
such
that
in
the
case
Qv
=
0
(v
=
1,
...,
4)
it
is
equal
to
the
constant
1
and,
upon
introduction
of
a
A-factor,
is
multiplied by
A-1.
I
would appreciate
hearing your opinion
on
these matters.-
Regarding
the
Nelson
topic,
I
thoroughly
understand
your
point
of
view.[6]
I
gladly
accept
the
prospect
of
discussing
it
with
you orally
when
the
occasion
presents
itself.-
Currently,
the
political
events
are
presumably
at
the
center
of
your
interest.
What
turn
destiny
will
take
“lies
in
the
lap
of
the
gods.”[7]
Most
cordial
regards,
yours,
Paul
Bernays.